Numbers, Operations, and Expressions
Review of Natural Numbers, Whole Numbers, Integers, and Rational Numbers
Classwork
1) Determine the classification(s) for each number below. List all that apply.
3
a)
11
b) –9.8
c) –21
30
d)
3
Review of Natural Numbers, Whole Numbers, Integers, and Rational Numbers
Homework
2) Determine the classification(s) for each number below. List all that apply
a) 2
3
b)
7
c) 0
72
d) −
6
e) 345.3
Review of Exponents, Squares and Square Roots
Classwork
3) Simplify
a) √121
b) 132
c) √225
d) 172
4) Determine the square root of each number. If the square root does not exist, write “no real solution”
121
a) √49
h) √−
256
b) √−25
196
c) −√289
i) −√
625
d) √−64
j) √0.64
e) √152
2
k) −√0.0144
f) −(√36)
l) √−0.0169
49
g) √
m) √3.24
144
5) Estimate each square root to the nearest integer
a) √39
b) √24
c) −√226
d) −√10
e) √130
f) −√292
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Review of Exponents, Squares and Square Roots
Homework
6) Simplify
a) √169
b) 192
c) √625
d) 122
7) Determine the square root of each number. If the square root does not exist, write “no real solution”
144
a) √−100
h) −√
289
b) √625
400
c) −√324
i) √−
81
d) √−36
j)
√0.25
2
e) √8
2
k) √−0.0064
f) −(√9)
l) −√0.0016
9
g) √
m) √2.25
676
8) Estimate each square root to the nearest integer
a) −√96
b) √37
c) √578
d) −√116
e) −√200
f) √411
Review of Irrational Numbers & Real Numbers
Classwork
9) Determine the classification(s) for each real number below. List all that apply.
a. √100
b. √15
4
c.
9
d. 0
e. –10.46
f. –11
g. 𝜋
21
h.
3
10) Determine whether each statement is true or false. Justify your answer.
a) The sum of two rational numbers is rational
b) The sum of a rational number and irrational number is rational.
c) The product of a nonzero rational number and an irrational number is irrational.
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Review of Irrational Numbers & Real Numbers
Homework
11) Determine the classification(s) for each real number below. List all that apply.
a) √65
b) −√25
c) 12
2
d)
5
e)
f)
g)
h)
√0
𝜋
−
6
12,385.93
–876
12) Determine whether each statement is true or false. Justify your answer.
a) The sum of two rational numbers is irrational
b) The sum of a rational number and irrational number is irrational.
c) The product of a nonzero rational number and an irrational number is rational.
Properties of Exponents
Classwork
13) Simplify each expression using the properties of exponents.
a) 𝑔7 ∙ 𝑔6
l) 𝑤 7 ÷ 𝑢−9
b) ℎ8 ÷ ℎ3
m) 𝑥 −4 ∙ 𝑦 7 ∙ 𝑧 −3
8𝑎4 𝑏 −5 𝑐 6
c) 𝑗 2 ∙ 𝑗 4
n)
d)
e)
f)
g)
h)
i)
j)
k)
𝑘7
𝑘2
o)
𝑥 5 ∙ 𝑥 11
𝑦 8 ÷ 𝑦10
90
7(20 )
8 + 30
11 − 3(60 )
𝑥 5 ∙ 𝑦 −8
p)
q)
r)
s)
t)
32𝑎−3 𝑏 2 𝑐 3
10𝑑 −2 𝑒 5 𝑓 −7
25𝑑 3 𝑒 −1 𝑓 −2
(𝑎4 )5
(𝑑 7 )4
(𝑏𝑐 3 )2
(2𝑒 2 𝑓 −3 𝑔5 )4
(
−2
8ℎ4 𝑗 5 𝑘 −3
3ℎ−2 𝑗 −3 𝑘
)
Properties of Exponents
Homework
14) Simplify each expression using the properties of exponents.
a) 𝑝4 ÷ 𝑝3
l) 𝑐 8 ÷ 𝑑 −10
7
4
b) 𝑞 ∙ 𝑞
m) 𝑒 5 ∙ 𝑓 −7 ∙ 𝑔4
c)
d)
e)
f)
g)
h)
i)
j)
k)
𝑟9
n)
𝑟3
3
𝑡 ∙ 𝑡4
𝑢5 ÷ 𝑢11
𝑣 8 ∙ 𝑣 10
50
13 + 90
8(40 )
15 + 4(70 )
𝑎−5 ∙ 𝑏 7
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o)
p)
q)
r)
s)
t)
~3~
9ℎ −4 𝑗 5 𝑘 −6
27ℎ3 𝑗 −2 𝑘 3
18𝑥 −1 𝑦 −5 𝑧 7
42𝑥 4 𝑦 −1 𝑧 −2
(𝑢3 )9
(𝑣 5 )6
(𝑎4 𝑏)3
(3𝑟 −2 𝑠 4 𝑡 2 )3
(
5𝑐 −3 𝑑 5 𝑒 3
7𝑐 4 𝑑 −2 𝑒
−2
)
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Like Terms
Classwork
Create a like term for the given term.
15) 4x
16) 13y
17) 15x2
18) 16xy
19) x
Simplify the expression if possible.
20) 7x + 8x
21) 6x + 8y + 2x
22) 15x2 + 5x2
23) 5x +2(x + 8)
24) -10y + 4y
25) 9(x + 5) + 7(x – 3)
26) 8 + (x – 4)2
27) 7y + 8x + 3y + 2x
28)
29)
30)
31)
32)
33)
34)
x + 2x
x2 + 5x2
2x + 4x + 3
6y – 3y
9y + 4y – 2y + y
x + 5x + x + 12
8x – 3x + 2x + 15
48)
49)
50)
51)
52)
53)
54)
x + 2x + x + 5x
6x2 + 5x2
12x + 14x + 3y
6y – 3y + 6xy + 4xy
9y + 4y – 2y + y + y2
x + 5x + x + 12 – 7x
8x – 3x + 2x + 15 – 7y
Like Terms
Homework
Create a like term for the given term.
35) 6x
36) Y
37) 10x2
38) 14xy
39) -5x
Simplify the expression if possible.
40) 17x + 18x + 3
41) 6x + 8y - 2x – y
42) 15x2 + 5x2 + 2x
43) 5x +2(x + 8) + 3
44) -10y + 4y – 5
45) 9(x - 5) + 7(x + 3)
46) 18 + (x – 4)2 – 4
47) 7y + 8x + 3y + 2x + 9
Evaluating Expressions
Classwork
Evaluate the expression for the given value
55) (2n + 1)2 for n = 3
56) 2(n + 1)2 for n = 4
57) 2n + 22 for n = 3
58) 4x + 3x for x = 5
59) 3(x – 3) for x = 7
60) 8(x + 5)(x – 2) for x = 4
61) 3x2 for x = 2
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62) 5x + 45 for x = 6
4𝑥
63)
for x = 10
5
64) 4y + x for x = 2 and y = 3
𝑥
65) + 17 for x = 12 and y = ½
𝑦
66) 6x + 8y for x = 9 and y = ¼
67) x + (2x – 8) for x = 10
68) 5(3x) + 8y for x = 2 and y = 10
Evaluating Expressions
Homework
Evaluate the expression for the given value
69) (2n + 1)2 for n = 1
70) 2(n + 1)2 for n = 3
71) 2n + 22 for n = 5
72) 4x + 3x for x = 6
73) 3(x – 3) for x = 3
74) 8(x + 5)(x – 2) for x = 6
75) 3x2 for x = 8
76) 5x + 45 for x = 3
4𝑥
77)
for x = 15
5
78) 4y + x for x = 12 and y = 13
𝑥
79) + 17 for x = 2 and y = ½
𝑦
80) 6x + 8y for x = 8 and y = ¾
81) x + (2x – 8) for x = 11
82) 5(3x) + 8y for x = 12 and y = 5
Ordering Expressions
Classwork
Order the terms by the degree of the variable in each expression.
83) 𝑥 + 12 − 4𝑥 3 − 5𝑥 2
84) 𝑤 2 + 10𝑤 − 8𝑤 3 − 3 + 5𝑤 4
85) 60 − 12𝑥𝑦 + 2𝑥 2 − 7𝑦 2
86) 34𝑢𝑣 − 8𝑢6 𝑣 5 + 42𝑢2 𝑣 2 − 52 − 15𝑢4 𝑣 4
87) 18𝑥𝑦 2 − 𝑥 3 + 81 − 7𝑥 2 𝑦 + 8𝑦 3
Ordering Expressions
Homework
Order the terms by the degree of the variable in each expression.
88) 11𝑥 2 − 4𝑥 + 17 − 3𝑥 3
89) 2𝑤 3 − 20𝑤 + 8𝑤 4 − 8 + 9𝑤 2
90) −13𝑝𝑞 − 19 + 3𝑝2 − 8𝑞 2
91) 36 − 14𝑢4 𝑣 3 + 23𝑢5 𝑣 4 − 54𝑢𝑣 − 5𝑢3 𝑣 2
92) 9 + 5𝑦 3 − 2𝑥 3 + 𝑥𝑦 2 − 20𝑥 2 𝑦
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Answer Key
1.
a.
b.
c.
d.
Rational
Rational
Rational, Integer
Rational, Integer, Whole, Natural
a.
b.
c.
d.
e.
Rational, Integer, Whole, Natural
Rational
Rational, Integer, Whole
Rational, Integer
Rational
a.
b.
c.
d.
11
169
15
289
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
7
no real solution
-17
no real solution
15
-36
a.
b.
c.
d.
e.
f.
6
5
-15
-3
11
-17
a.
b.
c.
d.
13
361
25
144
a.
b.
c.
d.
e.
f.
g.
no real solution
25
-18
no real solution
8
-9
j.
k.
l.
m.
0.5
no real solution
-0.04
1.5
a.
b.
c.
d.
e.
f.
-10
6
-24
-11
-14
20
a.
b.
c.
d.
e.
f.
g.
h.
Rational, Integer, Whole, Natural
Irrational
Rational
Rational, Integer, Whole
Rational
Rational, Integer
Irrational
Rational, Integer, Whole, Natural
8.
2.
3.
9.
4.
10.
a. True: If two rational numbers (or
fractions) are added together, then the
result has to be another rational number
3
5
18
(or fraction). For example, + = +
7
12
no real solution
14
−
25
0.8
-0.12
no real solution
1.8
20
24
38
24
=
19
12
4
𝑎𝑑+𝑏𝑐
6
24
, which is still a rational
𝑎
𝑐
𝑎𝑑
𝑏
𝑑
𝑏𝑑
number. In general, if + =
+
𝑏𝑐
𝑏𝑑
6.
11.
a.
b.
c.
d.
e.
f.
g.
h.
7.
3
Irrational
Rational, Integer
Rational, Integer, Whole, Natural
Rational
Rational, Integer, Whole
Irrational
Rational
Rational, Integer
12.
12
h. −
17
i. no real solution
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=
, where a, b, c and d are integers,
𝑏 ≠ 0, 𝑑 ≠ 0, then the sum is rational.
1
b. False: Counterexample = + 𝜋 cannot
2
be simplified. If you perform the sum
with the decimal equivalents, 0.5 +
3.14159… = 3.64159…
c. True: If a rational number, not equal to
0, and an irrational number are
multiplied together, the result has to be
irrational. For example, 2 ∙ √3 =
2(1.73205 … ) = 3.46410 …, which is still
irrational
𝑏𝑑
5.
26
=
~6~
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3
5
18
4
6
24
a. False: Counterexample: + =
20
38
19
+
m.
= = , which is a rational number,
24
24
12
not an irrational number.
b. True: If a rational number and irrational
number are added together, the result is
1
an irrational. For example, + 𝜋 cannot
2
be simplified. If you perform the sum
with the decimal equivalents, 0.5 +
3.14159… = 3.64159…, which is an
irrational number.
c. False: For example, 2 ∙ √3 =
2(1.73205 … ) = 3.46410 …, which is an
irrational number.
13.
a.
b.
c.
d.
e.
f.
𝑔13
ℎ5
𝑗6
𝑘5
𝑥 16
g.
h.
i.
j.
1
7
9
8
k.
l.
m.
n.
o.
1
𝑦2
𝑥5
𝑦8
9
𝑢 𝑤7
𝑦7
𝑥4𝑧 3
𝑎4 𝑐 3
4𝑏 7
2𝑒 6
5𝑑 5 𝑓 5
20
p. 𝑎
q. 𝑑 28
r. 𝑏 2 𝑐 6
s.
t.
16𝑒 8 𝑔20
𝑓 12
9𝑘 8
64ℎ12 𝑗 16
14.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
p
𝑞11
𝑟6
𝑡7
1
𝑢6
18
𝑣
1
14
8
19
𝑏7
𝑎5
8 10
𝑐 𝑑
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n.
o.
𝑒 5 𝑔4
𝑓7
𝑗7
3ℎ7 𝑘 9
3𝑧 9
7𝑥 5 𝑦 4
p. 𝑢27
q. 𝑣 30
r. 𝑎12 𝑏 3
s.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
27𝑠 12 𝑡 6
𝑟6
49𝑐 14
t.
25𝑑 14 𝑒 4
Multiple Answers ex:2(2x)
Multiple Answers ex:26y/2
Multiple Answers ex:(3x)(5x)
Multiple Answers (4x)(4y)
Multiple Answers ex:x2/x
15x
8x+8y
20x2
7x+16
-6y
16x+24
2x
10y+10x
3x
6x2
6x+3
3y
12y
7x+12
7x+15
Multiple Answers ex: 3(2x)
Multiple Answers ex. 5y – 4y
Multiple Answers ex. 5x(2x)
Multiple Answers ex. 7x(2y)
Multiple Answers ex. 5x – 10x
35x+3
4x+7y
20x2+2x
7x+19
-6y-5
16x-24
2x+6
10y+10x+9
9x
11x2
26x+3y
3y+10xy
12y+y2
12
7x+15-7y
49
50
10
35
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59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
12
144
12
75
8
14
41
56
22
110
9
32
14
42
0
352
192
60
12
64
21
54
25
220
NJ Center for Teaching and Learning
83.
84. −4𝑥 3 − 5𝑥 2 + 𝑥 + 12
85. 5𝑤 4 − 8𝑤 3 + 𝑤 2 + 10𝑤 − 3
86. 2𝑥 2 − 12𝑥𝑦 − 7𝑦 2 + 60 or
−7𝑦 2 − 12𝑥𝑦 + 2𝑥 2 + 60
87. −8𝑢6 𝑣 5 − 15𝑢4 𝑣 4 + 42𝑢2 𝑣 2 + 34𝑢𝑣 − 52
88. −𝑥 3 − 7𝑥 2 𝑦 + 18𝑥𝑦 2 + 8𝑦 3 + 81 or
8𝑦 3 + 18𝑥𝑦 2 − 7𝑥 2 𝑦 − 𝑥 3 + 81
89. −3𝑥 3 + 11𝑥 2 − 4𝑥 + 17
90. 8𝑤 4 + 2𝑤 3 + 9𝑤 2 − 20𝑤 − 8
91. 3𝑝2 − 13𝑝𝑞 − 8𝑞 2 − 19 or
−8𝑞 2 − 13𝑝𝑞 + 3𝑝2 − 19
92. 23𝑢5 𝑣 4 − 14𝑢4 𝑣 3 − 5𝑢3 𝑣 2 − 54𝑢𝑣 + 36
93. −2𝑥 3 − 20𝑥 2 𝑦 + 𝑥𝑦 2 + 5𝑦 3 + 9 or
5𝑦 3 + 𝑥𝑦 2 − 20𝑥 2 𝑦 − 2𝑥 3 + 9
~8~
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